Question

1 ∠3 ≅ ∠4 given
2 ∠1 and ∠3 form a linear pair∠2 and ∠4 form a linear pair definition of linear pair
3 ∠1 and ∠3 are supplementary∠2 and ∠4 are supplementary definition of supplementary angles
4 ∠1 ≅ ∠2 transitive property

Explanation:

Step1: Recall the definition of linear pair

A linear pair of angles is formed when two adjacent angles form a straight line. The sum of angles in a linear pair is \(180^{\circ}\).

Step2: Recall the definition of supplementary angles

Two angles are supplementary if the sum of their measures is \(180^{\circ}\). Since \(\angle1\) and \(\angle3\) form a linear pair, by the definition of supplementary angles, \(\angle1+\angle3 = 180^{\circ}\). Similarly, since \(\angle2\) and \(\angle4\) form a linear pair, \(\angle2+\angle4 = 180^{\circ}\).

Step3: Use the transitive property

We know that \(\angle3\cong\angle4\) (given). Let \(m\angle3=m\angle4\).
From \(\angle1+\angle3 = 180^{\circ}\), we can express \(m\angle1=180^{\circ}-m\angle3\).
From \(\angle2+\angle4 = 180^{\circ}\), we can express \(m\angle2=180^{\circ}-m\angle4\).
Since \(m\angle3 = m\angle4\), then \(m\angle1=m\angle2\). So, \(\angle1\cong\angle2\) by the transitive property.

Answer:

The table is correct. The reason for \(\angle1\) and \(\angle3\), \(\angle2\) and \(\angle4\) being supplementary is the definition of supplementary angles (as they form linear pairs). And the reason for \(\angle1\cong\angle2\) is the transitive property (since \(\angle3\cong\angle4\) and \(\angle1\) is supplementary to \(\angle3\), \(\angle2\) is supplementary to \(\angle4\)).